Manuscript Oai069-A@1 · Oai069-A.tex · lemma · a4913aab

Lemma: Encoding isometry and dual realization

Phase 2: positive realization and access-restricted recovery › The encoded local-system multiplicity · lines 676–693 · proof included or linked

Oai069-A@1#lem:encoding

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The map $V$ is isometric and preserves monodromy. Applying $F_{d}$ on the received $S$ gives

(F_{d} \otimes I) V : ∣ k ⟩ ∣ ψ ⟩ ∣ ξ ⟩ ⟼ ∣ k ⟩_{J} \otimes U_{k} (∣ ψ ⟩ ∣ 0 ⟩) \otimes ∣ ξ ⟩_{B} .

Proof. The $v_{k}$ are orthonormal and each $U_{k}$ is unitary. Therefore inner products are preserved. The monodromy and Fourier claims are Proposition prop:fourier.

Displayed formulas in this passage

unlabeled · line 680
$(F_{d} \otimes I) V : ∣ k ⟩ ∣ ψ ⟩ ∣ ξ ⟩ ⟼ ∣ k ⟩_{J} \otimes U_{k} (∣ ψ ⟩ ∣ 0 ⟩) \otimes ∣ ξ ⟩_{B} .$

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section: The encoded local-system multiplicity
section: The encoded local-system multiplicity